Thursday, April 18, 2013

Flying from Kepler-62 e to Kepler-62 f

Today NASA announced the discovery of two potentially-habitable planets orbiting a single star. The star is designated Kepler-62, and these two planets are referred to by their place in the five-planet system - e is planet four and f is planet five (a is reserved for the host star).

These planets are both quite small - 1.4 and 1.6 times the radius of the Earth, respectively. In addition, they're both located at relatively comfortable distances from their host star. Depending on what their atmospheres are like, they could potentially host conditions on their surfaces that are amenable to life.

But two planets in (or near) a single star's habitable zone? That made me wonder - if a civilization arose on one of the two planets, how difficult would it be for them to visit the other planet?

It's pretty easy to work out the change in speed needed to make the most efficient direct hop between two near-circular planetary orbits (as long as you're doing nothing fancy like gravitational assists, solar sails, or ion engines). I've illustrated what such an orbit (called a Hohmann Transfer Orbit) would look like between Kepler 62 e and f - this is very similar to the trajectory that we use to send our orbiters and landers from Earth to Mars.

This transfer orbit's semi-major axis (aship) is given by the average of the semi-major axes of the two planets (ae and af):

Once we have that, and we know a few things about the star the planets are orbiting, we can calculate the speed that a ship leaving planet e would need in order to reach planet f via this type of orbit. To get this, we can use the vis viva equation to find the orbital speed of a hypothetical ship on the transfer orbit when leaving planet e, then subtract off the orbital speed of planet e itself (since any ship launched would already be moving at the speed of the planet, and we're interested in how much more speed a rocket would need to give it):

Where G is the gravitational constant and M* is the mass of the host star.

So if we plug in these values (and the value of G with the correct units), we find a velocity of about 42.4 kilometers per second for the first term, and about 37.9 kilometers per second for the second term. Therefore, a rocket sending a ship from Kepler-62 e to Kepler-62 f would need to impart the difference, or about 4.5 kilometers per second.

Now we need some context - is 4.5 kilometers per second a lot? Let's compare it to the speed needed for a ship leaving Earth to reach Mars by the same type of orbit: all we have to do is substitute the Sun's mass for M*, Earth's semi-major axis (1 AU) for ae and Mars' semi-major axis (1.52 AU) for af. Plugging these in, we find about 32.7 kilometers per second for the first term and about 29.8 kilometers per second for the second term. So a rocket boosting a ship from Earth to Mars on this kind of transfer orbit would need to add about 2.9 kilometers per second.
So that means that a ship leaving Kepler-62 e for Kepler-62 f needs 55% more initial speed than a ship leaving Earth for Mars. Ouch.

But stay positive, you hypothetical planetary explorers from Kepler-62 e: just because it takes more oomph to get from Kepler-62 e to Kepler-62 f doesn't mean it's impractical! We've explored lots of places in our solar system that required more oomph than getting from Earth to Mars. In fact, NASA's Dawn spacecraft just finished exploring the asteroid Vesta (which orbits between Mars and Jupiter), and getting from Earth to Vesta on a Hohmann transfer orbit would require almost exactly the same initial speed as getting from Kepler-62 e to Kepler-62 f! Dawn took a very different trajectory, however; it rode out on low-thrust (but extremely efficient) ion engines, and it has now lit those engines up again to fly onward to the dwarf planet Ceres.